Is there a class of forcing notions which has the following property?
For every $A \subseteq \omega_1 \cap V[G]$, there exists $A' \subseteq \omega_1 \cap V$ with $|A \triangle A'| \leq \omega$? That is, new subsets of $\omega_1$ are "approximated" by ground model sets, modulo some countable set.
I'm not looking for something like $\omega_1$-closed forcings (that is any descending chain of $\omega_1$-many elements has a lower bound), as these would guarantee that new subsets of $\omega_1$ aren't added at all, etc.